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Cayley, Sylvester, and Early Matrix Theory
Nick Higham
School of Mathematics
The University of Manchester
http://www.ma.man.ac.uk/~higham/
SIAM Conference on Applied Linear Algebra
October 2009.
Cayley and Sylvester
Term “matrix” coined in 1850by James Joseph Sylvester,FRS (1814–1897).
Matrix algebra developed byArthur Cayley, FRS (1821–1895).Memoir on the Theory of Ma-
trices (1858).
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Biographies
Tony Crilly, Arthur Cayley: Mathemati-
cian Laureate of the Victorian Age,
2006.
Karen Hunger Parshall, James Joseph
Sylvester. Jewish Mathematician in a
Victorian World, 2006.
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Cayley SylvesterEnter Cambridge Uni-versity
Trinity College,1838
St. John’s College,1831
Wrangler in Tripos ex-aminations
Senior Wrangler,1842
Second wrangler,1837
Work in London Pupil barrister from1846; called to theBar in 1849
Actuary from 1844;pupil barrister from1846; called to theBar in 1850
Elected Fellow of theRoyal Society
1852 1839
President of the LondonMathematical Society
1868–1869 1866–1867
Awarded Royal SocietyCopley Medal
1882 1880
Awarded LMS De Mor-gan Medal
1884 1887
British Assoc. for theAdvancement of Sci-ence
President, 1883 Vice President,1863–1865; Presi-dent of Section A,1869
Academic Positions
Cayley Sylvester• Sadleirian Chair, Cam-bridge 1863
• UCL, 1838
• U Virginia, 1841• Royal Military Academy, Woolwich, Lon-don, 1855• Johns Hopkins University 1876• Savilian Chair of Geometry, Oxford,1883
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Crilly (2006):
Statistically, Cayley’s attention to matrix algebra is
even slighter than his attention to group theory and is
insignificant when compared to the large corpus he
produced on invariant theory.
Sylvester’s work was mainly algebraic.
Close friends: met around 1847.
Cayley: widely read, well aware of other research inBritain and continent.
Sylvester: mercurial, temperamental. Became involvedin a number of disputes:
“Explanation of the Coincidence of a Theorem Given by
Mr Sylvester in the December Nu mber of This Journal,
With One Stated by Professor Donkin in the June
Number of the Same”
Cayley’s Notation (1855)
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Cayley’s 1858 Memoir (1)
Introduces addition, multiplication, inversion, powering ofmatrices; zero and identity matrices.
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Cayley’s 1858 Memoir (2)
Cayley–Hamilton theorem: p(t) = det(tI − A) impliesp(A) = 0. Proved for n = 2:
I have not thought it necessary to undertake the
labour of a formal proof of the theorem in the
general case of a matrix of any degree.
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More General Cayley–Hamilton Theorem
Theorem (Cayley, 1857)
If A, B ∈ Cn×n, AB = BA, and f (x , y) = det(xA − yB) then
f (B, A) = 0.
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Cayleys Memoir: Matrix Functions
Finds some square roots of 2 × 2 matrices.And later in 1872:
X =A +
√
det(A) I√
trace(A) + 2√
det(A).
Finds parametrized family of 3 × 3 involutory matrices.
It is nevertheless possible to form the powers
(positive or negative, integral or fractional) of a
matrix, and thence to arrive at the notion of a
rational and integral function, or generally of
any algebraical function, of a matrix.
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Sylvester’s Words
annihilator, canonical form, discriminant, Hessian,Jacobian, minor, nullity.
Latent root:
It will be convenient to introduce here a notion(which plays a conspicuous part in my
new theory of multiple algebra),namely that of the latent roots of a matrix
—latent in a somewhat similar sense as vapour may besaid to be latent in water or smoke in a tobacco-leaf.
Derogatory, also called privileged!
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More Cayley Notation
Cayley introduced
AB−1 ≡A
B, B−1A ≡
A
B
and, in an 1860 letter to Sylvester,
AB−1 ≡A∼B
, B−1A ≡A∽
B.
Taber (1890) later suggested
A : B,A
B.
Hensel (1928) suggested A/B and B\A.MIMS Nick Higham Early Matrix Theory 13 / 18
Influence of the Memoir
Crilly (2006): that
Cayley’s ‘Memoir,’ which could have been a useful
starting point for further developments, went
largely ignored . . . His habit of instant publication
and not waiting for maturation had the effect of
making the idea available even if it was effectively
shelved.
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Sylvester on Matrix Theory
Sylvester worked on theory of matrices 1882–1884.
While teaching theory of substitutions, Sylvester
“lectured about three times, following the text
closely and stopping sharp at the end of the hour.
Then he began to think about matrices again. ‘I
must give one lecture a week on those,’ he said.
He could not confine himself to the hour, nor to the
one lecture a week. Two weeks were passed, and
Netto was forgotten entirely and never mentioned
again.”
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Sylvester’s Contributions
Investigated the matrix equation
AX 2 + BX + C = 0, A, B, C ∈ Cn×n.
in several papers published in the 1880s.
Sylvester’s law of inertia.
Sylvester equation AX + XB = C.
Gave first general definition of f (A).
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Recommended Reference
K. H. Parshall.
Joseph H. M. Wedderburn and the structure theory of
algebras.
Archive for History of Exact Sciences,
32(3-4):223–349, 1985.
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Matrices in Applied Mathematics
Frazer, Duncan & Collar, Aerodynamics Division ofNPL: aircraft flutter, matrix structural analysis.
Elementary Matrices & Some Applications to
Dynamics and Differential Equations, 1938.Emphasizes importance of eA.
Arthur Roderick Collar, FRS(1908–1986): “First book to treat
matrices as a branch of applied
mathematics”.
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References I
A. Buchheim.On the theory of matrices.Proc. London Math. Soc., 16:63–82, 1884.
A. Cayley.Remarques sur la notation des fonctions algébriques.J. Reine Angew. Math., pages 282–285, 1855.
A. Cayley.A memoir on the theory of matrices.Philos. Trans. Roy. Soc. London, 148:17–37, 1858.
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References II
A. Cayley.On the extraction of the square root of a matrix of thethird order.Proc. Roy. Soc. Edinburgh, 7:675–682, 1872.
T. Crilly.Cayley’s anticipation of a generalised Cayley–Hamiltontheorem.Historia Mathematica, 5:211–219, 1978.
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References III
T. Crilly.The appearance of set operators in Cayley’s grouptheory.Notices of the South African Mathematical Society, 31:9–22, 2000.
T. Crilly.Arthur Cayley: Mathematician Laureate of the Victorian
Age.Johns Hopkins University Press, Baltimore, MD, USA,2006.ISBN 0-8018-8011-4.xxi+610 pp.
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References IV
R. A. Frazer, W. J. Duncan, and A. R. Collar.Elementary Matrices and Some Applications to
Dynamics and Differential Equations.Cambridge University Press, Cambridge, UK, 1938.xviii+416 pp.1963 printing.
K. Hensel.Über den Zusammenhang zwischen den Systemen undihren Determinanten.J. Reine Angew. Math., 159(4):246–254, 1928.
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References V
N. J. Higham.Functions of Matrices: Theory and Computation.Society for Industrial and Applied Mathematics,Philadelphia, PA, USA, 2008.ISBN 978-0-898716-46-7.xx+425 pp.
I. Kleiner.A History of Abstract Algebra.Birkhäuser, Boston, MA, USA, 2007.ISBN 978-0-8176-4684-4.xiii+168 pp.
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References VI
B. W. Levinger.The square root of a 2 × 2 matrix.Math. Mag., 53(4):222–224, 1980.
K. H. Parshall.Joseph H. M. Wedderburn and the structure theory ofalgebras.Archive for History of Exact Sciences, 32(3-4):223–349,1985.
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References VII
K. H. Parshall.James Joseph Sylvester. Life and Work in Letters.Oxford University Press, 1998.ISBN 0-19-850391-1.xv+321 pp.
K. H. Parshall.James Joseph Sylvester. Jewish Mathematician in a
Victorian World.Johns Hopkins University Press, Baltimore, MD, USA,2006.ISBN 0-8018-8291-5.xiii+461 pp.
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References VIII
J. J. Sylvester.Explanation of the coincidence of a theorem given by MrSylvester in the December number of this journal, withone stated by Professor Donkin in the June number ofthe same.Philosophical Magazine, (Fourth Series) 1:44–46, 1851.
Reprinted in [18, pp. 217–218].
J. J. Sylvester.On Hamilton’s quadratic equation and the generalunilateral equation in matrices.Philosophical Magazine, 18:454–458, 1884.Reprinted in [19, pp. 231–235].
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References IX
The Collected Mathematical Papers of James Joseph
Sylvester, volume 1 (1837–1853).Cambridge University Press, 1904.xii+650 pp.
The Collected Mathematical Papers of James Joseph
Sylvester, volume 4 (1882–1897).Chelsea, New York, 1973.xxxvii+756 pp.Corrected reprint, published in four volumes, of workpublished in four volumes by Cambridge UniversityPress 1904–1912.
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References X
H. Taber.On the theory of matrices.Amer. J. Math., 12(4):337–396, 1890.
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